What is a Rhombus?
Definition: A rhombus is a closed two-dimensional shape and it satisfies all properties of a parallelogram. But it has a unique identity due to some special properties. A parallelogram is a quadrilateral in which the opposite sides are parallel. In addition to this condition, all sides of a rhombus are equal and the diagonals of a rhombus intersect each other at 90 degrees.
Rhombuses, rhombi, are referred to as diamonds due to their resemblance to the shape of a diamond. A rhombus is also known as an equilateral quadrilateral, as the sides of a rhombus are equal. Rhombuses are also quite similar to squares as both shapes have four equal slides. But a rhombus can look like a slanted square as the four angles of a rhombus need not be 90 degrees at all times.
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Properties of Rhombuses
- All sides of a rhombus are equal and parallel to each other.
Rhombus
- The opposite angles of a rhombus are equal.
- A rhombus cannot be a cyclic quadrilateral, meaning it cannot be inscribed inside a circle, unless it is a square.
- The diagonals of a rhombus bisect each other at right angles.
- The sum of all interior angles of a rhombus is always 360 degrees.
- The sum of the adjacent angles of a rhombus is always 180 degrees (supplementary).
- The diagonals of a rhombus bisect its angles.
- Perimeter of a rhombus = 4a units, where a is the length of each side.
- Area of rhombus \( =\frac{d_1\times d_2}{2} \) square units, where \( d_1 \) and \( d_2 \) are its diagonals.
- If the length of the shorter diagonal of a rhombus is equal to the length of its side, the diagonal divides the rhombus into two equilateral triangles.
Relationship with other Shapes
Quadrilaterals are broadly categorized into three families: kites, parallelograms and trapezoids. A rhombus is a special type of quadrilateral. It is a kite and a parallelogram at the same time. We can relate a rhombus to other shapes as follows:
- All rhombuses are parallelograms but not all parallelograms are rhombuses.
- All rhombuses are kites but not all kites are rhombuses.
- All squares are rhombuses but not all rhombuses are squares.
Solved Rhombus Examples
Example 1: Determine whether the following shape is a rhombus.
Solution:
All four sides of this quadrilateral are parallel to each other. Hence, it is a parallelogram.
Also, the four sides are equal in length and the diagonals bisect each other at right angles. Hence, this quadrilateral is a rhombus.
Example 2: Find the perimeter of the rhombus.
Solution:
We know that the given shape is a rhombus. All four sides of a rhombus are of equal length. Therefore,
Perimeter of rhombus = 4a
= 4 \( \times \) 15
= 60 inches
Therefore, the perimeter of the rhombus is 60 inches.
Example 3: Liza and Brian are making two kites on their own. Both kites have the shape of a rhombus. The diagonals of Liza’s kite are 3 feet and 4 feet long and the diagonals of Brian’s kite are 2 feet and 6 feet long. Who has the bigger kite?
Solution:
We can determine who has the bigger kite by finding the area of both kites.
Length of the diagonals of Liza’s kite: 3 feet and 4 feet
Length of the diagonals of Brian’s kite: 2 feet and 6 feet
Area of a rhombus \( =\frac{d_1 \times d_2}{2} \)
Area of Liza’s kite \( =\frac{3 \times 4}{2} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=3 \times 2 \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=6 \) square feet.
Area of Brian’s kite \( =\frac{2 \times 6}{2} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=2 \times 3 \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=6 \) square feet.
So, they have kites of the same size.
By definition, a rhombus is a shape that has equal sides that are parallel to each other. Since a square fits this definition, we can say that all squares are rhombus. But the converse is not always true.
Since the opposite sides of a rhombus are parallel to each other and equal in length, we can consider a rhombus a parallelogram. But the converse is not always true. For a parallelogram to be a rhombus, its sides need to be of the same length.
A kite is a shape in which there are two pairs of adjacent sides of equal length. Since the adjacent sides of a rhombus are equal, we can say that all rhombuses are kites. But the converse is not always true.
Rhombuses and squares share some properties. All sides of a square and rhombus are equal in length and the diagonals bisect each other at right angles. But unlike rhombuses, all four angles of a square measure 90 degrees. The difference in the angles makes rhombuses appear like slanted squares.